Issue sorted.
Over the weekend, I tried two different approaches.
First, I got the bike up on the centre stand and removed, inspected and cleaned off both wheel-speed sensors (thinking that perhaps dirt, debris, metal particles etc. might have been interfering with the readings forwarded by one or both sensors).
I picked up nothing of particular interest, except that the front sensor was so close to the disc that it had in fact been scraping it at one point. (I couldn't get a 0.05mm feeler-gauge blade between sensor and disc - the Haynes manual specifies an air gap of between 0.2mm and 0.7mm.) I spaced out the front sensor to 0.6mm using a thin 4mm flat washer.
It made no difference to the ASC malfunction. But there was a moral: Never assume that the guy who built your bike built it properly.
With the sensors now a known quantity, I pulled the rear wheel off my K1200S streetfighter and exchanged the worn-out 190/50-17 tyre for an ex-track-day 190/55-17 Pirelli Diablo Supercorsa I picked up for next to nothing, mounted the 'wrong' way round on the rim (on K-series, the rear wheel bolts on from the right-hand side).
From the time the new combo went on, the ASC worked perfectly. (I'm now getting a tyre-pressure readout error, but that's natural - the streetfighter's rear wheel has no pressure sensor.)
I can only assume that the 180/55-17 tyre the bike was wearing when it was sold to me was only just inside the 'acceptable' parameters for rolling circumference (taking into account a combination of R1200S gear ratios and GS Adventure ASC software), and as it started to wear, the circumference must have eroded enough to cause the fault to manifest. Turns out, the extra rolling circumference given by the taller tyre profile was the 'magic bullet' I needed.
And, I was really surprised at how well the 190/55 complements the GSA.
The original 180/55 (a Michelin Pilot Road 3) gave stable, neutral handling when I pushed the bike hard. But with the Pirelli, it's just more, more, more. The bike turns in much more easily, is calmer in fast bends, and holds it's line more accurately. There is also an improvement in ride comfort.
Make no mistake, the original wheel/tyre configuration was good. But with the 'new' config, the bike feels like it's just exchanged Wellies for track running spikes.
A 190/55-17 Pilot Road 3 is definitely on the cards.